内容来源：https://www.quantamagazine.org/what-does-the-fourth-dimension-actually-look-like-20261001/

内容总结：

四维空间究竟什么样？数学家带你一探究竟

我们熟知长、宽、高三个维度构成的空间。物理学家常把时间称为第四维度，但如果存在第四个空间维度——一个全新的方向——我们该如何想象它？

数学家借助拓扑学来 visualize 高维抽象空间。德克萨斯大学奥斯汀分校的玛吉·米勒（Maggie Miller）专门研究当纽结遇到额外维度时会发生什么。在四维空间中，熟悉的缠绕环圈会表现出出人意料的复杂和反直觉行为：纽结总能被解开，而球面等普通曲面反而会以无法解开的方式打结。

在 Quanta Magazine 的播客节目《The Joy of Why》中，米勒向联合主持人詹娜·莱文（Janna Levin）解释了为什么四维是数学家至今仍未完全理解的最低维度，她和合作者如何解决了数学家查尔斯·利文斯顿（Charles Livingston）于1982年提出的关于打结曲面的问题，以及她对艺术的兴趣如何帮助她发展出用于想象四维空间的视觉技术。

拓扑学：研究“橡皮泥几何”的学问

拓扑学研究抽象空间，关注的是物体在连续变形下保持不变的性质——可以拉伸、扭曲，但不能撕裂或穿透。这与我们熟悉的几何学不同：几何学关心距离和角度，而拓扑学关心的是全局连通性。一个圆和一个椭圆在拓扑学家眼中是同一个对象，但圆和直线则截然不同。

米勒将拓扑学描述为“万物皆由橡皮制成”的学科。她以莫比乌斯带为例：沿中线剪开一个莫比乌斯带，它不会分成两半，而是变成一个更大的环——这一反直觉的性质正是拓扑学关注的核心。

从三维到四维：直觉为何失效

米勒指出，三维拓扑与四维拓扑之间存在巨大鸿沟。在三维中，我们凭借日常经验可以做出许多正确猜测；但在四维中，直觉几乎总是错误的。她用“2+2=4”这一看似无用的说法来解释原因：在四维空间中，一个二维圆盘可能与自身相交于一点，而无法被推开，因为相交处已经占满了全部四个维度。这一现象导致许多在三维中成立的定理在四维中失效。

四维中的纽结：一切都能解开

在三维空间中，纽结是打结的闭合环圈，无法在不切断的情况下解开。但米勒解释说，一旦进入四维空间，所有纽结都会变得平凡——它们全部等价于一个简单的圆环。这是因为多出的一个维度提供了足够的“活动空间”来解开任何缠绕。然而，对于四维中的“曲面纽结”（即二维曲面在四维空间中的打结方式），情况则复杂得多，至今仍是活跃的研究领域。

塞弗特曲面与利文斯顿问题

米勒与凯尔·海登、金成完、朴正焕和艾萨克·桑德伯格合作，解决了利文斯顿在1982年提出的问题。利文斯顿证明了：对于平凡链环（一组互不缠绕的圆环），如果两个连通塞弗特曲面具有相同的亏格，那么在四维空间中它们可以互相变形。他猜测，如果边界不是平凡链环，这一结论就不成立，但未能证明。

米勒团队成功构造出反例，证明了利文斯顿的猜想。这一成果需要先构造出复杂的高维曲面实例，再证明它们在四维中无法互相变形——这正是拓扑学中最困难的部分。

数学与艺术：相通的视觉思维

米勒从小热爱艺术，大学时在数学和艺术之间选择了数学。她认为两者本质相通：解数学题和画画都是解谜过程，都需要视觉思维和技术规划。她将四维空间画成一系列三维“切片”的序列，就像电影的帧一样。她甚至有一个“秘密目标”：画出六维或七维空间的图像，其中一个维度通过翻页来体现。

四维的特殊性

米勒指出，二维曲面可以被完全分类，三维空间也有完整的分类定理（由瑟斯顿和佩雷尔曼完成），但四维空间既无法分类，又充满反常现象，因此成为数学家最感兴趣也最困惑的维度。更高维度虽然更加抽象，但反而具有某些统一的规律。

纯粹数学的价值

对于“这些研究有什么用”的问题，米勒坦言，纯粹数学与应用之间通常有约50年的时间差。拓扑学中较新的“拓扑数据分析”领域正在将经典拓扑工具应用于大数据分析，但当代拓扑学家正在做的研究，可能还需要几十年才能找到实际应用。她幽默地提到，拓扑学家发明新工具的方式之一，就是“走到走廊对面的物理系，请他们给出一个新的偏微分方程”，然后带回数学系，忘记它的物理来源，将其作为区分空间的新方法。

工作中的快乐

米勒表示，最大的快乐来自解决难题的那一刻——“就像填字游戏终于找到答案时的豁然开朗”。她形容自己一直在“能力边缘”工作，有时非常沮丧，但每次突破都带来巨大的满足感。

中文翻译：

第四维空间究竟是什么样子的？

引言

我们都熟知事物如何在长、宽、高这三个维度中存在。物理学家常常将时间称为第四维，但假如存在第四个空间维度——一个完全不同的方向呢？我们究竟该如何想象它？

数学家利用拓扑学来可视化高维抽象空间。德克萨斯大学奥斯汀分校的玛吉·米勒探索了当纽结遭遇额外维度时会发生什么。熟悉的缠绕环圈在四维空间中表现出意想不到的复杂且反直觉的行为：纽结总是可以解开，而诸如球面这样的普通曲面则会——令人惊讶地——以无法解开的方式打结。

在本期节目中，米勒向联合主持人詹娜·莱文解释了为什么四维是数学家至今仍未完全理解的最低维度，她和合作者如何解决了数学家查尔斯·利文斯顿于1982年首次提出的关于打结曲面的问题，以及她对艺术的兴趣如何帮助她发展出用来想象四维空间的可视化技法。

可在Apple Podcasts、Spotify、TuneIn或您喜爱的播客应用上收听，也可以从Quanta网站流媒体播放。

文字记录

[音乐播放]

詹娜·莱文：我们开始了。我是詹娜·莱文。

史蒂夫·斯特罗加茨：我是史蒂夫·斯特罗加茨。

莱文：这里是《为什么之乐》。

斯特罗加茨：Quanta杂志出品的一档播客，我们在这里探索当今数学和科学中一些最重大的未解之谜。

莱文：史蒂夫，我们今天的话题非常数学化。我很高兴有一位数学家跟我同谋。

斯特罗加茨：好，听您差遣。

莱文：太好了。在本期节目中，我们要谈论的是数学的一个分支，叫做拓扑学，很多人对它并不熟悉。

你能不能帮我们了解一下拓扑学的基本知识？拓扑学和我们可能学过的普通几何学有什么不同？

斯特罗加茨：当然可以。拓扑学，非常热门的学科。你知道，几何学历史悠久，而拓扑学——取决于你从什么时候算起——可能只有几百年历史，但它真正兴起是在20世纪。

人们可能在童年游戏中接触过它，比如你有没有拿过一张纸，像一条长条、某种丝带，然后扭一下，扭半圈，再把两端接起来，用胶带粘上，做出一种叫做莫比乌斯带或莫比乌斯环的形状？你有没有跟你的孩子一起做过？

莱文：没有，但我对莫比乌斯带的思考频率大概已经到不健康的程度了。

斯特罗加茨：哦，真的吗？

莱文：是的。它出现在我的研究中。是的。

斯特罗加茨：哦，好吧。有一次我的一个孩子上小学，正好是“带家长上学日”，我就让孩子们做莫比乌斯带，然后做那件事——沿着带子的中线一路剪下去，绕整圈剪开。你以为你在把它剪成两半，但它并不会散开。然后一个孩子就开始哭了。

莱文：哦[笑]。好吧，数学确实让人沮丧。

斯特罗加茨：就是这样，这是对所有数学家和数学老师所做之事的一种早期入门体验——把人弄哭在数学课上。

莱文：其实这很完美。[笑]

斯特罗加茨：好吧。总之，这就是我们的主题——拓扑学。我是说，莫比乌斯带，这个性质——当你沿中线剪开时，它不会分成两块——这是一个关于事物如何连接的拓扑性质。你可以改变很多几何性质，但不会改变这个性质。它比几何更深层。

莱文：没错。我们通常想到几何学时，会想到勾股定理，它精确地告诉我如何测量距离，而拓扑学不是关于这个的。它是关于整体的连通性。所以它具有非常抽象的特征，而我们今天的嘉宾正是玛吉·米勒，德克萨斯大学奥斯汀分校的助理教授，她研究纽结、曲面和四维空间。

那么，废话不多说，让我们听听玛吉·米勒对拓扑学的看法。

[音乐播放]

莱文：欢迎来到《为什么之乐》，玛吉。非常高兴你能来。

玛吉·米勒：谢谢邀请我。我很兴奋能来这里。

莱文：我对你的工作非常着迷。它我个人也很感兴趣。我有很多问题想问你，但让我们先从玛丽亚姆·米尔扎哈尼新前沿奖开始吧，它属于突破奖的范畴。你几年前获得了这个奖，你提到在大学早期，你在数学和艺术之间犹豫不决，我觉得这是一个非常迷人的组合。是什么特别的原因让你最终选择了数学？

米勒：从小时候起，甚至在上大学之前，我就非常投入于艺术。我上了很长时间的美术课。

我是说，我知道很多人在学校里上美术课。这是一门非常常见的受欢迎科目，但我一直参加我们州的州立展览。这就是我真正喜欢的东西，那种视觉方面的思考、规划和实际制作某物的技术层面。

我上大学时，其实立刻就选择了数学。因为数学一直是我作为学生最喜欢的科目。我想说在学校里，但我其实是在家上学的，所以这有点不太清楚是什么意思。

但它在某种意义上让我觉得和艺术很相似，都是一个小谜题。我想这在数学上更明显。也许人们能理解我说的解数学题就像解谜题是什么意思，但我对画画也有同样的感觉。我是说，任何曾经为画面比例挣扎过的人都知道，要弄清楚东西到底该放在哪里并不总是那么容易。

莱文：我同意。这是艺术中一个被误解的方面。人们认为艺术没有结构、没有约束，但事实并非如此，尤其是如果艺术家很优秀的话。

米勒：没错。

莱文：所以我得听听关于在家上学的事。

米勒：是的。我父母决定让我和我双胞胎姐姐从三年级读到一半左右开始在家上学。这在德克萨斯州非常普遍。我成长过程中认识很多其他在家上学的孩子。我有两个比我小很多的弟弟，所以他们两个一直在家上学，直到上大学，这again，在德克萨斯州比其他一些州更普遍，我觉得在政府限制方面也稍微宽松一些。

这也有它自己的挑战，因为并不总是很清楚你应该做什么。尤其是作为青少年，我知道我想上大学。当时，德克萨斯大学实际上有一个网页，上面列出了他们对在家上学的学生可能被录取的期望。所以我在大概14岁的时候看到了那个，我和妈妈一起列了一个清单——这些是所有大致等同的东西。什么算作历史？

莱文：所以，玛吉，每次我问你一个问题，你都会说一些让我不得不追问的事。先是关于在家上学，然后是“我和我的双胞胎姐姐”。双胞胎姐姐，一个同卵双胞胎姐姐，她也是数学家或艺术家吗？

米勒：不，我们是异卵的。我觉得我们有很多共同点。我们仍然是姐妹。她对传统意义上的数学或艺术不是特别感兴趣。她肯定有很多她喜欢的手工艺活动。我姐姐上大学学的是心理学，所以我觉得她对社会科学可能更感兴趣一些。

莱文：嗯嗯。是的。那么，我想进入讨论你的数学领域，这是我最喜欢的学科之一：拓扑学。你能解释一下什么是拓扑学吗？

米勒：拓扑学是研究抽象空间的学科，这可能意味着我们现在所生活的空间。我们生活在一个我们感知到三个维度的空间里，X、Y和Z。但它指的是一个更一般的概念。什么是空间，这是你在拓扑学课程中一开始就要定义的东西。但现在，就想象一个你可以居住的空间。我们说两个物体或空间是相同的、等价的，如果你能把一个连续变形为另一个。所以拓扑学研究的是拉伸和扭曲物体的学问，但你永远不能弄断东西。

莱文：所以当你说X、Y和Z时，你指的是我们习惯占据的三个维度。但在拓扑学中，你也研究其他维度的空间。

米勒：没错。研究任意维度的空间或没有良定义维度的空间是非常常见的，因为它们可以非常抽象。举个例子，一个非常简单的空间例子是圆。不是实心圆，而只是圆盘的边缘，就像我拿一个盘子的边缘之类的。

所以这是一个一维物体的例子。如果我站在一个圆上，那么我只有一个方向可以走，就是顺时针或逆时针，这是同一个方向但方向相反。但我可以构造一个非常复杂的空间，比如如果我拿一百万个圆，把它们全都在一个点上粘在一起呢？

如果我拿无限多个圆——然后我们得考虑是哪种无限——把它们全都在一个点上粘在一起。如果我做这样复杂的事情，整体理解这个物体就难多了。

莱文：所以如果你把它们全都在一个点上粘在一起，它几乎就像一个扇子。

米勒：是的，会是一个扇子，是的。或者有时人们叫它耳环。

莱文：所以有些空间是名副其实的“空间”。它们代表着我们可以生活的地方，或者可以被占据的地方，但有些空间与我们习惯的物理维度无关。

米勒：没错。我觉得物理维度激发了这个名字，这是拓扑学和数学其他领域中的一个常见主题。有一个简单的例子，它有一个合理的英文名字，然后我们把它类比推广，仍然使用同一个名字，即使感觉上那不是我们在英文意义上的意思。

莱文：嗯嗯。那么，为了获得某种视角，让我们从三维流形的概念开始。我们大概生活在一个三维流形中。你能给我们讲讲我们拥有三个维度是什么意思吗？

米勒：好的。当我们谈论维度时，它指的是有多少个不同的垂直方向，使得每个方向都是这些方向的组合。所以我们生活在一个三维空间中，因为有三个垂直方向：前、后、左、右、上、下。它们彼此之间都是90度。

我想去的任何方向都是这三个方向的某种组合。它们也占满了它们的维度，所以这意味着它恰好是一个三维空间。

莱文：所以，如果人们不习惯思考90度和正交性，我喜欢的一种理解方式是：如果某个方向与其他方向正交，我可以在一个维度上移动而完全不在其他方向上移动。

米勒：没错。我是说，如果我想告诉你我确切在哪里，那需要多少条不同的信息？我需要给你我在这三个方向上的坐标。

棘手的是，首先我们都得同意在.000处意味着什么。所以我们称之为原点，然后从那里我可以测量——嗯，我在那个方向上离零点有多远？然后前后多远，然后上下多远。

另一个棘手的事情是，对于拓扑学家来说，我们其实没有距离的概念。所以我说我们可以测量我离原点有多远时，有点误导，因为如果我是一个拓扑学家，我其实不知道我离任何一个点有多远。我只知道两个点是否相同。没有“更远”意味着什么的概念。

莱文：嗯嗯。你关心的是事物是否可以在一个空间和另一个空间之间平滑变形。所以你不关心它是否是橡胶做的，我是否真的拉伸或压缩了距离。你关心的是，如果我走得足够远，我会不会回到起点？某种整体性质。这样说公平吗？

米勒：完全正确。是的，所以我们谈论局部结构与整体结构。对于拓扑学家来说，一切都是橡胶做的，只要我能拉伸两个物体让它们看起来一样，那它们就是一样的。所以圆就是椭圆。但它们绝对和直线不一样。

莱文：对。因为直线不是以那种特定方式连接的。所以你说拓扑学是研究那种大的、整体的结构。现在，你特别在四维空间中工作过，你能用时间的类比给我们讲讲第四个方向吗？因为我觉得这能帮助人们想象当我们思考一个超出我们习惯占据的三个空间维度之外的维度时，我们在想什么。

米勒：没错。拥有第四个维度是一个真的很抽象的概念。人们理解前、后、左、右、上、下。这是三个正交的垂直方向。

所以，如果我们把这三个方向视为代表空间——我们生活的通常的三维空间——我们必须思考，在一个不同的方向上移动意味着什么，这意味着我的空间坐标不变，所以我们可以把它想成时间。

所以现在，我在时间中以——我猜是以每秒向前一秒的速度在时间中前进。

但我在三维空间中的位置没有改变。所以这意味着我可以认为我现在生活在一个四维空间中，其中第四维是时间，它垂直于前三个维度。这有点像一部电影，如果我在看电影，我可以倒带或暂停。

莱文：我们如何从三维移动到四维？

米勒：嗯，我认为四维拓扑学的一个主题是，你不能轻易地从三维拓扑学过渡到四维拓扑学。

所以在三维中，我们对什么应该是真的有很强的直觉，因为我们生活在一个看起来是三维的空间中。这不一定对超级抽象的问题有帮助，但至少对于简单的行为，你大概可以根据你的生活经验猜到会发生什么。

但四维不同，很难概念化。我有很多论文的格式是：“这是三维中成立的东西，这是它在四维中不成立的例子。”我觉得这是一个很有趣的探索。

在拓扑学中有很多定理或事实，由于某种原因在其他维度中成立，但在四维中不成立，或者也许我们在其他维度中理解了答案，但在四维中还没有。关键原因往往是——我知道这是一个你不喜欢听到的答案——关键原因往往是二加二等于四，说起来这真是毫无帮助。

但问题是，拓扑学中有一些定理，其证明经过这样一步：“我在我的抽象空间中有一条曲线或一个环，我想把它缩成一个点。”

比如我有一个很大的圆，我想让那个圆的半径随时间趋于零，而曲线不断裂。一个问题是，假设我能做到。我做到的一种方式是，我不仅找到一个圆，而是找到一个完整的圆盘，整个填充区域，因为那会给我如何实际缩小圆的指令。我只需要说：“哦，我把那个圆盘缩到它的中心点”，它就会把圆也一起带到中心点。

圆盘是二维的，因为二加二等于四，这告诉我，如果我们在四维空间中，这个圆盘实际上可能与自身相交。它会在点处与自身相交。因为不知怎么的，如果我有四个维度——让我们叫它们X方向、Y方向、Z方向，现在还有一个W方向，所以我有四种不同的垂直方式可以走。我有某个点，它在圆盘中，圆盘向X和Y方向延伸。但在自相交处，有另一块圆盘，它向Z和W方向延伸。

所以我没办法把圆盘从自身推开并消除那个交点，因为它占据了全部四个维度。这阻止了我一定能缩小曲线，而它在拓扑学中一直出现。这就像是证明中的一个关键问题。

莱文：所以，我听你说过我们不应该如此依赖直觉，因为当我们试图把我们非常常识性的直觉从三维转移到四维时，我们会感到非常惊讶。它完全欺骗了我们。

米勒：是的。我喜欢告诉学生们有三门非常重要的课，就是实分析、代数结构和拓扑学。实分析本质上是微积分的延续，大多数上那门课的学生可能从微积分经验中有很多直觉，而且他们的直觉大概是对的，他们在学习如何将其形式化。当你学习代数结构时，你可能根本没有太多直觉，因为它非常抽象。

但上拓扑学的学生有直觉，因为我们在谈论看起来熟悉的物体，我们可以画很多图，而直觉总是错的。很容易被困住。

莱文：那么，你用什么技巧来避免被困住？

米勒：[笑]嗯，如果我实际上试图通过图像来理解物体，如果我实际上在视觉上构造一个物体，那么我需要有一个严格的体系，让图像真正有意义。它不能只基于我对物体行为的直觉。我需要形式化——如果我画一个平面和一条与之相交的直线，那到底意味着什么？所以这里涉及到一些实分析。

所以我们都得同意这些图示到底意味着什么，然后我们还得制定规则——这是我可以对空间或空间中纽结的图像做的事情的规则，做了改变之后，图像仍然描述同一个空间或物体或纽结。所以一旦我完成了这些工作，理解了这些图示的分析，那就给了我更多的灵活性来实际做构造，并知道我没有意外地丢失信息。

莱文：所以，这是视觉表示和数学形式化的结合？

米勒：没错，并不是所有事情都通过视觉表示来完成。我喜欢那样工作，因为那是真正吸引我进入数学的东西，但有很多优秀的拓扑学家，你知道，四维拓扑学家，他们在工作中真的不使用视觉表示。它并不总是重要的。

但我确实想让人们理解，即使数学中有很多图，你会经常从学生那里听到这个，这不是用图来证明。这不是我们随便编的一个图。这里有很多——没错——形式化的工作融入到这些论证中。

莱文：有了这些四维空间和你的直觉失灵，你问的是什么样的问題？你真正在追求的是什么假设？

米勒：我最常研究的是打结曲面。在三维中，类似的物体是纽结。纽结——k-n-o-t。纽结就是一个环。如果我拿一根绳子……等等，我其实有一根绳子。如果我拿一根绳子，我真的把它缠起来，然后我把两端粘在一起做成一个闭合的环，那我就叫它纽结。

莱文：对。

米勒：是的，我们研究这些物体，是在连续变形的意义下。所以我可以拉伸纽结、扭曲它，但我永远不能让它穿过自身。这是三维拓扑学中的一个重要对象。

它为什么重要不是显而易见的。需要一些理论来理解纽结为什么重要，但事实证明，每一个三维空间都可以用经典空间内部的打结环圈来理解。

所以在四维中，我研究打结曲面，就是所有升一维的东西。不是环，而是二维的东西。

莱文：像一条丝带。

米勒：嗯，不太像丝带，更像一个气球。我想要一个没有边缘的物体。但也有更复杂的曲面。气球是一个球面或2-球面。说空间的维度是很常见的。

它是二维的，因为如果我站在一个气球上，我可以在气球上前后左右移动，但如果我试图向上走，我就会离开气球。也许不说气球，我应该说是地球表面，另一个二维球面的例子。如果我向上进入太空，我就离开了地球表面，所以它只是二维的。

但还有更多的空间，比如环面，如果一颗小行星撞击地球并直接穿透，我就会得到那个二维空间。所以现在上面有个洞。它是那个的表面。我觉得更正常的说法是它是百吉饼或甜甜圈的表面。然后如果我有多几个洞，我还可以得到更多的空间。

所以我可以研究嵌入或打结在四维空间中的曲面。所以想象起来有点难，但有一个版本的缠绕——一个2-球面在四维空间内部。

莱文：我有好几个问题。一个是你说过你可以用这些纽结来理解所有三维空间。

米勒：没错。

莱文：这似乎是一个很大的飞跃。你能帮我们建立这个联系吗？

米勒：这是一个大定理，我想是1960年代由利科里什和沃利斯提出的，利科里什-沃利斯定理。更具体地说，我研究的是三维流形，我想说这些三维流形没有边界，它们是紧致的，意思是我永远不会走到无穷远。所以我可以——也许我会说有限体积。我还希望它们是可定向的。所以想想莫比乌斯带作为一个不可定向的例子，因为人们常说它只有一个面。所以我加了一些限制。我只想研究合理的三维空间。

但一旦我这样做了，利科里什和沃利斯证明了，我可以从标准的那个——叫做3-球面——出发得到任何我想要的三维空间。它像R³，但如果我试图走到无穷远，最终我会回到我出发的点。所以我不能永远走下去。

我所要做的就是取一些纽结——3-球面内部的一些打结环圈——然后对它们做手术。这让人困惑。这是拓扑学中一个非常重要的操作。我觉得第一次听到它时，你会惊讶它居然能做到任何事情。但当我说手术时，我的意思是我要删除那些纽结。所以我在删除这些打结的环圈。我想把它们想成有厚度的。就像我从空间中切出一些百吉饼。然后我要重新粘合，或者精确地替换我删除的东西。所以我切出一些这样的加厚圆圈，这些实心百吉饼，实心环面。然后重新接上它们。但我要以一种不同于我发现它们的方式重新接上，因此我会改变空间。

莱文：我明白了。所以如果我理解正确的话，纽结本身——你说我可以从单连通的有限空间，也就是3-球面出发，然后通过对纽结进行这类手术来构造三维中的任何其他拓扑。你是这个意思吗？

米勒：没错，完全正确。这叫做德恩手术。

莱文：所以这样你就制造出越来越多复杂空间，远远超出光滑、紧致、连通的三维球面的简单性。你制造出这些带有把手和洞的非常复杂的空间，有点像折纸。

那么现在带我升到四维。现在我们试图想象纽结在四维中的角色，但纽结不再是绳子。它们本身就是曲面，复杂的曲面。

米勒：没错。

莱文：而你试图做同样的事情。你试图从四维中最简单的空间出发，在纽结的基础上找到所有其他的？

米勒：是的。现在几乎可以说我可以从最简单的空间出发，叫做4-球面，它更难想象因为它是四维的。但它是某种标准的四维空间，我们都同意从它出发。我要取的曲面是环面，就是这些百吉饼或甜甜圈的表面坐在四维空间内部。我在做手术，意思是我一次删除一个。我要删除一个环面，重新粘回去，但我可以用一种有趣的方式重新粘，然后我会改变四维空间。

所以我说几乎可以说我可以得到任何东西。有两个非常基本的数值不变量，我必须从正确的值出发。所以至少有一个非常简单的四维空间族，我从其中出发。取决于我想得到什么空间，我知道从哪个开始，我所要做的就是对接环面做手术，我就能得到我想要的任何空间。

莱文：我明白了。这是一个成功的纲领吗？

米勒：嗯，这是关于四维的一个事实。这是一个很酷的定理，我想是岩濑几年前做的。它不完全是纲领。它更多是一个动机，说明为什么研究曲面是值得的。任何关于四维空间的问题，原则上我都可以重新表述为关于曲面的问题，而这只是一个例子。四维空间还有其他方面，我们倾向于用曲面来研究。

比如，让我们回到低维空间。让我们再看看曲面。我之前给了两个曲面的例子，2-球面和环面。第一次学习这些物体时，真的很难用语言表达它们之间的区别。当我做一个面向更广泛听众的演讲时，我会说：“嗯，环面上有个洞，球面上没有洞。”但如果我问一个学生：“好，现在告诉我什么是洞。”——环面有个洞是什么意思？这是一个棘手的问题。

我喜欢这样想：如果我在2-球面上画一个环，比如想想地球的赤道，它总是会把2-球面分成两块。赤道把地球分成北半球和南半球，你不能不穿过赤道就在它们之间移动。

环面没有这个性质，因为我可以在环面上取一个绕着洞的环，如果我切，它实际上不会把环面切成两块。但我只能放一个这样的环。如果我试着沿两个环切，那我最终会把环面切成两块。所以，这告诉我环面有一个洞，我可以用这些子空间——这些维度更小的、坐在曲面内部的圆——来形式化它。所以，我也可以在高维中这样做。如果我试图区分两个4-流形，有时我可以把它重新表述为关于“什么样的曲面坐在这些四维空间内部”的问题。

[音乐播放]

斯特罗加茨：所以当我戴着拓扑学家的帽子听这些时，我想到的是玛吉在这里使用的一个策略，我们被训练在拓扑学中一直使用它。就是像她说的那样，用这些更低维的物体来探测更高维的物体。因为这不是我们每天都会做的事，我想与其谈论四维的用法，不如我们就谈谈像厨房餐桌这样的东西？

莱文：嗯嗯。

斯特罗加茨：你知道，比如如果我在桌上放一根绳子，把它弄成圆的形状，我一这样做就发生了一个非常深刻的拓扑学事件，那就是绳子的环把桌子分成了两部分，一部分我们叫里面，一部分我们叫外面。

这挺有意思的。你可能会说：“不，不是的。这太明显了。”但这就是数学家思考的那类事情。比如，好，让我更抽象一点。不是桌面，而是无限平面，你知道，向两个方向延伸到无穷的XY平面？

如果你在XY平面中放一个圆，它把平面分成一个有界的内部部分和一个包含无穷的外部部分，即使我变形这个环，这仍然成立，总是有一个里面和一个外面。

莱文：是的。你看，我们有皮肤，二维曲面，它在三维中把我们的内部和外部隔开。但如果我们在四维中生活，它就不会了。我们的皮肤不足以完全包裹我们的内脏。

斯特罗加茨：那么我们不住在四维中就很方便了，因为我们会被洒出来。

莱文：对。我们会——你就可以直接进去摘出来，不用做手术。

斯特罗加茨：手术会容易得多。

莱文：是的。她也谈到了类似你课堂经历的东西，就是某种意义上，空间的连通性可以通过思考你如何分割事物来分类，本质上你需要切多少刀才能真正把东西切成两半。

斯特罗加茨：对。她讨论中另一个触动我的点是，说清楚“洞”是什么意思有多难。你知道，这在互联网上真正成为热议话题的地方是——如果我看一条裤子，就是普通的裤子，上面有几个洞？

莱文：好，有一个，两个。

斯特罗加茨：但有些人想说三个，因为你知道，你一条腿底部有一个环，另一条腿上也有，然后还有腰带。看起来好像有三个。

莱文：嗯，好吧。等等。用可收缩的环来说，我有一个，两个……我猜……三个。

斯特罗加茨：我要让听众自己思考这个问题。你接下来要带我们去哪里？

莱文：好的。史蒂夫，有件事我想请你帮忙。广告之后有一个话题会出现，那就是赛弗特曲面。我想你思考过这个。

斯特罗加茨：我思考过。它们实际上在我的一些应用数学工作中出现过。也许一个简单的想象方式是，想象你小时候可能玩过的一个游戏，你拿一个金属丝环来吹泡泡。然后如果你把它浸入肥皂水中，你会得到一种肥皂膜，横跨那个环。那个圆盘就是赛弗特曲面的一个例子，因为它是一个曲面，其边界是一条给定的曲线，在这个例子中是一个环。

但你也可以有打结曲线的赛弗特曲面，其中那个曲面的边界是纽结。这在纽结理论中是一个非常有用的构造。

莱文：这确实是一个有用的可视化，我们将在广告之后深入讨论这个以及更多内容。

[音乐播放]

莱文：欢迎回到《为什么之乐》。今天我们的嘉宾是玛吉·米勒，德克萨斯大学奥斯汀分校的拓扑学家。

让我们谈谈你回答查尔斯·利文斯顿1982年提出的一个长期问题的这项工作，你和你的合作者完成的，受到了很多关注，它与赛弗特曲面有关。那些曲面有什么特别之处？

米勒：是的，没错。我应该说那是与凯尔·海登、金成完、朴正焕和艾萨克·桑德伯格一起做的。赛弗特曲面不仅仅是一个抽象曲面，它是一个实际生活在三维空间内部的曲面。所以我真的想做出这个区分，尽管我知道甜甜圈生活在我们三维空间内部。我把它当作自己的抽象对象，就像在空间之外。它在某个奇怪的地方。但赛弗特曲面真的在这里。

它们生活在三维空间内部这一点很重要，因为当我研究这个曲面时，我可以连续变形它，但它必须留在三维空间内部。所以我可能有两个不同的环面，它们都是环面，但我实际上不能在三维空间内部把一个变成另一个。

更简单的事情就是纽结，打结的环。如果我取一个环，它实际上就是一个圆，一个漂亮的圆，我们叫它平凡纽结。如果我取一个环，真的把它打成某种复杂的东西，它就不是平凡纽结了。它们仍然都是圆，但它们不是同一个纽结。我实际上不能在不让它穿过自身的情况下把一个变成另一个。赛弗特曲面也是一样。

但对于赛弗特曲面，我确实希望它们有边界。它们应该有一个边缘，边缘将是一个纽结，一个打结的环。

莱文：所以这个想法是你在想象，至少对某些纽结来说，如果你增加它们所生活的维度数量，你也许能在不切割、不断裂、不粘合的情况下把其中一个解开成另一个。

米勒：没错。这经常发生。所以就纽结而言，比如一个打结的环，它打结只是因为我们处于三维空间中。如果我有第四个维度的自由，如果我进入四维空间，那么我的纽结实际上会变得平凡。与周围空间相比，它的维度太小了，不可能再复杂。

莱文：所以它会有效地解开？它会有效地变成平凡纽结？是的。

米勒：没错。它就会直接解开。所以在四维中，在标准的四维空间中，恰好只有一个打结的环。它们都是一样的。

莱文：啊，我明白了。但如果它们不是……等等，你是说如果三维中有纽结，你把维度升到四维，那它们就都是一样的了？

米勒：那它们就都是一样的了。

莱文：但如果你有一个生活在四维中的纽结，所以它是一个曲面纽结，你再升一个维度，它们也都会变得一样吗？

米勒：那就不太清楚了，但如果我升两个维度到六维，那它们就都是一样的了。如果我增加足够多的维度，最终它们都会变得一样。

莱文：所以当我学了一点这个的时候，你可以取一个二维的例子，可能更容易让人可视化。如果你想象桌面上一个小橡皮筋在一个大橡皮筋里面。你没法让它们互相穿过，对吧？

但如果你有三维，就很容易了。你把一个拿起来，拉伸它，然后你就可以把它放在另一个外面。所以这个想法——维度的数量允许你在不切割和粘贴的情况下解开东西或分开东西——是有点直观的。我们一直在用它。

米勒：没错。这正是正确的想法。这和为什么三维中的纽结在四维中变得平凡或都一样是同一个道理。就像，我只是需要那一个额外的维度来拉橡皮筋。

莱文：那么让我们回到赛弗特曲面。你解决的关于它们在四维中行为的问题是什么？

米勒：是的。我们真的对查尔斯·利文斯顿——查克·利文斯顿——的一篇论文很感兴趣，他在研究平凡纽结和分离链的赛弗特曲面，其中分离链是一堆彼此不相连的平凡纽结。就是一堆彼此无关的圆。是的。所以尽管分离链在某种意义上不是很有趣，我仍然可以找到复杂的、彼此不同的分离链的赛弗特曲面。所以曲面边界是平凡的、不有趣的。但曲面本身是复杂的。

利文斯顿证明了，如果我有两个——比如说——同一个分离链的连通赛弗特曲面，并且我知道这两个曲面有相同的亏格，亏格是我们衡量曲面有多少个洞的指标。

尽管那些曲面在三维空间内部可能不一样，如果我加一个第四维度的自由，它们就变得一样了。我有这个额外的方向可以把它们拉过去。在论文中，他很具体地说明边界是分离链，他指出如果边界不是分离链，这肯定不成立。但他没有证明，主要是因为想出高维物体的奇怪例子真的很难。你首先得想出一对具有相同边界的疯狂曲面，然后才能试图证明它们彼此不同。这在拓扑学中一直是个问题。

莱文：是的。我觉得我们可以再多解释一下“这些是一样的”这个想法。这确实是拓扑学的核心原则之一——你说的“一样”是什么意思？你能澄清一下吗？

米勒：是的，这是一个核心原则——两个物体是一样的意味着什么。这也是一个核心原则——有很多不同的“一样”的概念。所以这总是——每当你定义一个对象时，你必须说，两个对象一样将意味着什么。到目前为止我们谈论了流形，包括曲面和纽结。对于流形，我们通常使用同胚的定义。

莱文：嗯嗯。

米勒：我们说两个流形是同胚的，如果存在从一个到另一个的同胚映射。这意味着，如果我有两个流形，叫它们X和Y，它们是同胚的，如果我能找到一个函数F，从X到Y。这个函数需要是连续的，还需要是双射，意味着它是一对一且满射的。

所以X中的任意两个点必须映射到Y中的两个不同的点。它们不能映射到同一个点。我还必须覆盖Y中的每一个点。X中的某个东西必须映射到它。最后一件事是我希望f也有连续的逆。所以这只是一组从X到Y的连续指令，但这三个要点使它成为一个严格的定义。

所以在纽结的语境中，我其实可以使用同样的定义。我可以说两个纽结是等价的，如果存在一个同胚——这种受限的函数——从3-空间到自身，但当我代入第一个纽结时，它输出第二个纽结。

莱文：如果我给一个拿着橡皮泥的孩子这个形式化的数学描述，我会说：“游戏是，你能把这个形状变成那个形状而不弄断它，不戳破也不重新粘合吗？”只要你只是在平滑地揉捏橡皮泥，那么在拓扑学上我们就认为它们是同胚的。

米勒：对。是的。所以拉伸东西是可以的。扭曲东西是可以的。如果我有两个纽结，如果我有一个纽结，我试图把它变成另一个纽结，我可以把绳子变长，我可以加扭曲，但我永远不能让绳子穿过自身，那会弄断它。

莱文：现在，我觉得游戏的问题并非完全不相关，因为对于数学之外的人来说，当他们听数学家正在解决的问题时，会有一种试图评估的倾向——这只是一个被玩的游戏吗？

我觉得用哲学方式表述这个问题就是：你认为数学是被发现的东西，它在某种意义上是存在于那里的，还是数学是人类发明的一系列不同的游戏？

米勒：哦，这很有趣。我觉得，嗯，我觉得最根本或最重要的结果是被发现的东西，但我们到达它们的方式往往是通过赢得一个游戏。我觉得这完全正确。如果我试图解决一个问题，比如说我有一个我认为是真的猜想，我需要证明它对所有纽结都成立。那么我可能把它表述为一个游戏——好吧，如果我给你一张纽结的图片，我告诉你：“好，你可以从这张图片开始，但你可以用以下方式改变图片”，我确切告诉你什么是允许的，是否可能只做那些事情并最终得到我猜想的那个东西？

就像，发现的部分，我觉得，是首先想出这个游戏。做出猜想真的很难，因为我们不总是有好的直觉。然后很多论文就是——好吧，这里有一个我们都感兴趣的大问题，我的定理是我实际上不能回答它，但我能证明如果你总能赢得这个游戏，那么定理就是真的。

莱文：对于数学之外的人，他们试图理解关键在哪里，为什么这些不同的游戏重要且有吸引力？我知道这对数学来说是一个存在性的问题。

米勒：我们之前提到的游戏总是归结为——好吧，如果我能赢得这个游戏，那我就证明了这个猜想。所以从根本上说，这真的是为什么纯数学重要的问题。一个问题是，对于在这个领域工作的任何人来说，它为什么对现实世界重要并不总是清楚的。

人们喜欢开玩笑说，纯数学被发明或发现——或者你想用的任何动词——与实际应用到现实生活中之间大概有50年的时间差。比如，在拓扑学中，有一个相对较新的领域叫拓扑数据分析，你从名字就能立刻看出它对现实生活更适用。

它的想法是我们如何利用拓扑学来高效地理解非常大的数据集或理解其中的一些模式。所以也许我有某个数据集，它太大了，我没法搜索，但我可以理解全局模式，比如，这个数据集我可以用这个空间来近似，这是有意义的，而它是这个空间这个事实告诉了我一些关于数据的信息。

但实际进入拓扑数据分析的拓扑学——纯数学部分——在这一点上，就纯粹的数学组成部分而言，相对来说是旧的。也许我们在使用曲面的分类或拓扑学中的一些经典不变量，所以它在某种程度上仍在追赶。拥有那些东西很重要，但我们这周在拓扑学中做的事情还没有被应用到应用数学中。

莱文：是的。我理解在某种意义上是这是一个不公平的问题。它更多是给数学家一个机会来回应它，因为物理学家一直面临类似的问题。你的工作对我的日常生活有什么意义？真正的答案是那不是我们关心的，而那没关系。

米勒：嗯是的，真正的答案是那不是目标。事实上，我觉得大多数拓扑学家，如果你问他们，他们可能会试图把它联系到物理学，然后希望你不要问物理学那部分为什么重要。

莱文：对。你知道吗？让我们玩那个游戏，因为那很有趣也很有意思。在一种非常有价值的意义上，拓扑学对理论物理学有巨大的应用。拓扑学有很多复杂的应用方式，但有一个是有点直观的，就是宇宙可能是一个三维流形。如果我们把时间算进去，它可能是一个四维流形，对吧？不只是另一个空间方向，而是时间，在那个流形上时间的度量与空间的度量不同。你会思考这些对物理学的意义吗？

米勒：嗯，实际上有趣的是，我觉得大多数拓扑学家会往相反的方向想，然后说：“嗯，我从不思考物理学家需要什么，抱歉。”但我对物理学家带给拓扑学的东西非常感兴趣。

所以当我读研究生时，一位给我上课的教授开了这个玩笑，我至今一直在想，完全正确。每当拓扑学家想要一种区分两个空间的新方法时，他们做的就是走过走廊到物理系——肯定在同一栋楼里——问他们：“哦，你能给我一个新的偏微分方程，告诉我一些关于空间的信息吗？”然后你把它带回来，完全忘了它和物理学有任何关系，那就是你的新方法。是的，所以我们喜欢说：“哦，物理学家告诉我有一个描述热流的偏微分方程，我可以数有多少个稳定解之类的，我就从现在开始记住那个数字。我不会去想热流，但它对这两个空间是不同的，所以它们一定不同。”

莱文：我喜欢这个。我喜欢它是双向的。当然，物理学也必须面对为什么是三个维度的问题。所以很多物理学家对高维空间感兴趣，因为我们认为宇宙很可能是更高维的。

所以当我们试图理解什么是可能的宇宙，可能宇宙的几何是什么时，我们必须求助于数学家，说：“好吧，如果我们在四维中，这些在某种意义上是唯一的宇宙，而这些尽管看起来可能不同，实际上是相同的。”这当然与我们要做的事情非常相关。

米勒：是的。这完全合理。当然，有很多数学家研究高维空间。很多人会说：“哦，我是三维流形拓扑学家。我是四维流形拓扑学家。”具体到更高的数字就不太常见了。比如，没人会说他们研究八维拓扑学……

莱文：不会吗？为什么？因为超过八维就太可概括了，就只是N了？

米勒：它们往往是三维特有的、四维特有的技术，然后是更高的。但有很多人研究更高维度，尤其是代数拓扑学往往研究更高的任意维度。

莱文：出于好奇，三维和四维有什么特别之处吗？

米勒：是的，有特别之处，我觉得三维在某些方面类似于二维。

在二维中，曲面我可以枚举。我可以完全列出所有不同的二维物体，你真正需要给我的数据并不多，就能完全确定我拥有的是哪个二维流形。

在三维中，陈述起来稍微复杂一些，但有一个三维物体的完全分类，这要现代得多。这大概是从1970年代到2000年代，瑟斯顿和佩雷尔曼的工作。

我们完全……好吧，也许我想对“完全”这个词小心一点，但我可以说我们完全理解三维流形。至少我们有一个完整的刻画。还有很多有趣的问题，但一般不是“这两个空间是否不同”的问题。

在四维中，那是一个更难的问题，但有很多兴趣，不一定是分类。那是不可能的。但它很特殊，因为来自更高或更低维度的定理

英文来源：

What Does the Fourth Dimension Actually Look Like?
Introduction
We know all about how things exist in the three dimensions of length, width, and height. Physicists often talk about time as a fourth dimension, but what if there were a fourth spatial dimension — another direction entirely? How on earth would we picture that?
Mathematicians use topology to visualize abstract spaces in higher dimensions. Maggie Miller at the University of Texas at Austin explores what happens when knots encounter an extra dimension. Familiar tangled loops behave in unexpectedly complex and counterintuitive ways in 4D: Knots can always come undone, while ordinary surfaces like spheres can — surprisingly — become knotted in ways that can’t be undone.
In this episode, Miller explains to co-host Janna Levin why 4D is the lowest dimension that mathematicians still don’t fully understand, how she and her collaborators resolved a question about knotted surfaces first posed by mathematician Charles Livingston in 1982, and how her interest in art has helped her develop the visual techniques she uses to picture 4D spaces.
Listen on Apple Podcasts, Spotify, TuneIn or your favorite podcasting app, or you can stream it from Quanta.
Transcript
[Music plays]
JANNA LEVIN: And we’re off. I’m Janna Levin.
STEVE STROGATZ: And I’m Steve Strogatz.
LEVIN: And this is The Joy of Why.
STROGATZ: A podcast from Quanta Magazine where we explore some of the biggest unanswered questions in math and science today.
LEVIN: Steve, we have a really mathematical topic today. I’m glad to have a mathematician in cahoots.
STROGATZ: Okay, at your service.
LEVIN: Excellent. We’re speaking in this episode about a branch of mathematics known as topology, and that’s not familiar to a lot of people.
Do you wanna help us a little bit just with the basics of topology? What makes topology different than regular geometry that we might have learned?
STROGATZ: Yeah, sure. Topology, very popular subject. You know, whereas geometry is ancient, topology is only, depending where you date it, could be a few hundred years old, but it really took off in the 20th century.
People may have run into it in childhood games, like did you ever take a piece of paper, like a strip, some kind of ribbon, and then you put a twist in it, half a twist, and then close it, put a piece of tape on it to make this shape called a Möbius strip or a Möbius band? Did you ever do it with your kids?
LEVIN: No, I didn’t, but I think about the Möbius band probably an unhealthy amount.
STROGATZ: Oh, really?
LEVIN: Yeah. It comes up in my work. Yeah.
STROGATZ: Oh, okay. Well, there was a time when one of my kids was in elementary school, and it was bring your parent to school day, and so I had the kids make Möbius strips, and then do this thing where you cut down the midline of the strip all the way around the whole ring. And you think you’re cutting it in half, but it doesn’t fall apart. And so one kid started crying.
LEVIN: Oh [laughs]. Well, math is frustrating.
STROGATZ: This is the thing, it was an early introduction to what all mathematicians and math teachers are trying to do, make people cry in math class.
LEVIN: It’s perfect, actually. [laughs]
STROGATZ: Okay. But anyway, so that’s our subject, topology. I mean, the Möbius strip, this property that when you cut it down the middle, it doesn’t fall into two pieces, that’s a topological property about how things are connected. You could change a lot of geometrical things, and you wouldn’t change this property. It’s deeper than geometry.
LEVIN: Right. Well, we usually think about geometry, we think about the Pythagorean theorem, which is precisely telling me how to measure distances, and topology is not about that. It’s about this global connectedness. So it has this very abstract character, and our guest today is, in fact, Maggie Miller, an assistant professor at the University of Texas at Austin, who researches knots, surfaces, and four-dimensional spaces.
So, uh, without further ado, let’s hear what Maggie Miller has to say about topology.
[Music plays]
LEVIN: Welcome to The Joy of Why, Maggie. It’s really a pleasure to have you on.
MAGGIE MILLER: Yeah, thanks for having me. I’m excited to be here
LEVIN: I’m really fascinated by your work. It intrigues me personally. There’s a lot of questions I wanna ask you, but let’s just start with the Maryam Mirzakhani New Frontiers Prize, which is within the Breakthrough Prize umbrella. You received that a few years ago, and you mentioned that early on in college, you were split between studying math and art, which I think is a really fascinating combination. What in particular led you to choose math in the end?
MILLER: As a kid, even before college, I was really invested in art. I took art classes for a long time.
I mean, I know lots of people take art in school. It’s a very common favorite subject, but I always participated in our state fair. It’s just something I really like, that visual aspect of thinking and planning and the actual technical aspect of making something.
When I went to college, actually, I chose to do math right away. And it’s because math had always been my favorite subject as a student. I wanna say in school, but I was actually homeschooled, so it’s a little bit unclear what that means.
But it actually felt similar to me to art in terms of it’s a little puzzle. I guess it’s more apparent with math. Maybe people can understand what I mean when I say solving a math problem is like a puzzle, but I feel like that about drawing something too. I mean, anybody who’s struggled with aspect ratios knows it’s not always so easy to figure out where things are actually supposed to go.
LEVIN: I agree. This is sort of a misunderstood aspect of art. People think art is structureless and without constraints, and that’s really not the case, especially if the artist is good.
MILLER: That’s right.
LEVIN: So I need to hear about the homeschooling.
MILLER: Yeah. My parents decided to homeschool me and my twin sister, starting about halfway through third grade. It is very common in the state of Texas. I knew a lot of other homeschooled children growing up. I have two brothers who are quite a bit younger than me, and so they were both homeschooled all the way, up until college, which again, it’s very common in Texas compared to some other states, and I think it’s a little easier in terms of government restrictions.
That produces its own challenges, because it’s not always so clear what you’re supposed to be doing. Especially as a teenager, I knew I wanted to go to university. At the time, the University of Texas actually had a webpage that had a list of what they expected from homeschoolers in order to potentially be accepted. And so I saw that when I was around 14, and I worked with my mom to make a list of, here’s all the things that are roughly equivalent. What counts as history?
LEVIN: So, Maggie, each time I ask you a question, you say something that I have to ask about. So first it was the homeschooling, and then it was me and my twin sister. Twin sister, an identical twin sister, and is she also a mathematician or an artist?
MILLER: No, we’re fraternal. I think we have a lot in common. We’re still sisters. She is not particularly interested in math or art in the traditional sense. She definitely has a lot of craft activities that she enjoys. My sister went to college and studied psychology, so I think she’s a little bit more interested in maybe social science.
LEVIN: Mm-hmm. Yeah. So, I wanna get into discussing your area of mathematics, which is one of my favorite subjects: topology. Can you explain what topology is?
MILLER: So topology is the study of abstract spaces, which could mean the space that we live in right now. We live in a space where we perceive three dimensions, X, Y, and Z. But it refers to a more general concept. There’s what is a space is something that you start off defining in a course on topology. But for now, just imagine, like, a space you could live in. and we say that two objects or spaces are the same, equivalent if you can continuously deform one into the other. So topology is the study of objects like stretching and twisting, but you’re never allowed to break things.
LEVIN: So when you say X, Y, and Z, you mean the three dimensions we’re used to occupying. But within topology, you also work on other dimensional spaces.
MILLER: That’s right. It’s very common to study spaces of any dimension or spaces that don’t have a well-defined dimension because they can be really abstract. So for example, a very simple example of a space is something like the circle. So not the filled-in circle, but really just the edge of a disc, like if I took the edge of a plate or something.
So that’s an example of a one-dimensional object. If I’m standing on a circle, then there’s one direction I can go, which is clockwise or counterclockwise, which is the same direction but backwards. But then I could construct a really complicated space where what if I just took a million circles and glued them all together at one point?
What if I took an infinite number of circles, and then we have to consider what kind of infinity, and I glued them all together at one point. And if I do something complicated like that, it’s a lot harder to understand the object in whole.
LEVIN: So if you glue them all together at one point, it’s like a fan almost.
MILLER: Yeah, it would be a fan, yeah. Or sometimes people call it an earring.
LEVIN: So some of these spaces are true to the word “space.” They represent, where we can live, or places that can be occupied, but some of them don’t have to do with that physical dimensionality that we’re used to.
MILLER: That’s right. I think that the physical dimensionality motivates the name, and that’s a common theme in topology and other areas of math. There’s one easy example which has an English name that makes sense, and then we analogize it and still use the same name even when it feels like that’s not what we mean in the English sense.
LEVIN: Mm-hmm. So, to gain some kind of perspective, let’s start with the idea of a three-dimensional manifold. We live presumably in a three-dimensional manifold. Can you walk us through the sense in which we have three dimensions?
MILLER: Yeah. So when we talk about dimension, it means how many different perpendicular directions are there so that every direction is a combination of those. So we live in a three-dimensional space because there are three perpendicular directions, forward, backward, left, right, up, down. Those are all 90 degrees away from each other.
Any direction that I would want to go is some combination of those three. So they take up all of their dimension as well, so that means it’s exactly a three-dimensional space.
LEVIN: So, if people aren’t used to thinking about 90 degrees and orthogonality, one way I like to think of it is I can move in one dimension if it’s orthogonal to the others without moving in the other directions at all.
MILLER: That’s right. I mean, it’s, if I wanted to tell you exactly where I was, it would be how many different pieces of information do I need to tell you? So I need to give you my coordinates in those three directions.
The tricky thing is first we have to all agree what does it mean to be at .000? So what is the, we call it the origin, and then from there I can measure, well, how far in that direction am I from zero? And then how far forward, backward, and then how far up, down.
Another tricky thing is that for topologists, we actually don’t have a notion of distance. So, I was being a little bit misleading when I said we could measure how far away I am from the origin, because if I’m a topologist, then I actually don’t know how far I am from any one point. I just know whether or not two points are the same. There’s no notion of what it means to be farther away.
LEVIN: Mm-hmm. What you care about is if things are smoothly deformable between one space and another space. So you don’t care if it’s made of rubber and I literally stretch the distances or contract the distances. What you care about is, if I walk far enough, will I come back to where I started? Some kind of global property. Would that be fair?
MILLER: That’s exactly right. Yeah, so we talk about local structure versus global structure. For a topologist, everything is made of rubber, and as long as I can stretch two objects and have them come out looking the same, then they are the same. So a circle is an oval. But they’re definitely not the same thing as a line.
LEVIN: Right. ’Cause a line isn’t connected in that particular way. So topology you’re saying is that study of that large, that global structure. Now, you’ve in particular worked in four dimensions, can you talk us through the fourth direction using the time analogy? Because I think that helps people kind of imagine what we’re thinking about when we’re thinking about a dimension beyond the three spatial ones we’re used to occupying.
MILLER: That’s right. I mean, having a fourth dimension is a really abstract idea. People understand forward, backward, left, right, up, down. Those are three orthogonal perpendicular directions.
So, if we’re thinking of those three directions as representing space, the usual three-dimensional space we live in, we have to think what would it mean to move but in a different direction which means that my spatial coordinate does not change, and so we can think of that as time.
So right now, I’m moving forward in time at a rate of, I guess I’m moving one second forward in time per second.
But my position in three-dimensional space is not changing. So that means that I could think that right now I’m living in a four-dimensional space where that fourth dimension is time, and it’s perpendicular to the first three. It’s a little bit more like a movie, where if I were watching a movie, then I could rewind or pause.
LEVIN: How do we move to dimension four from dimension three?
MILLER: Well, one theme, I think, of four-dimensional topology is that you can’t easily move from three to four-dimensional topology.
So in dimension three, we have a lot of intuition for what should be true, because we live in a space that seems to be three-dimensional. That doesn’t necessarily help with super-abstract problems, but at least simple behavior, you can probably guess what will happen just based on your lived experience.
But dimension four is different and hard to conceptualize. I have a lot of papers where the format of the paper is, “Here’s something that’s true in dimension three, and here’s an example of how it’s not true in dimension four.” I think that’s a fun thing to explore.
There are many theorems or facts in topology that for some reason hold in other dimensions, but not in dimension four, or perhaps we understand the answer in other dimensions, but not in dimension four. The key reason tends to be, and I know this is like an answer that you hate to hear, but the key reason tends to be that two plus two equals four, which is such an unhelpful thing to say.
But the problem is that there are some theorems in topology where the proof goes through a step of, “I have a curve or a loop inside of my abstract space, and I want to shrink it to a point.”
So I have like a really big circle, and I want to over time just have that radius of that circle go to zero without the curve breaking. And one issue is that let’s suppose that I could do that. One way that I would do that is I would find not just a circle, but a whole disc, so the whole like filled-in region, because that would give me instructions for how to actually shrink the circle. I would just say, “Oh, I’ll shrink that disc down to its center point,” and it’ll bring the circle down to the center point with it.
So a disc is two-dimensional, and because two plus two is equal to four, that tells me that it’s possible that this disc actually intersects itself if we’re in four-dimensional space. It would intersect itself in points. Because somehow if I have four dimensions, so let’s call them the X direction, Y direction, Z direction, and now a W direction, so I have four different perpendicular ways I can go. I have some point where it’s in the disc and the disc goes off in the X and Y directions. But at the self-intersection, there’s another piece of the disc and it goes off in the Z and W directions.
And so there’s no way I could push the disc off of itself and get rid of that intersection because it’s taking up the whole four dimensions. That stops me from necessarily being able to shrink curves, and it just shows up all the time in topology. It’s like a key problem in the proof.
LEVIN: So, I’ve heard you say also that we shouldn’t be so reliant on our intuition ’cause we can be incredibly surprised when we try to transfer our, our really common-sense intuition from three dimensions to four dimensions. It just completely deceives us.
MILLER: Yeah. I like to tell students there are three really important classes that students take, which are real analysis, algebraic structures, and topology. Real analysis is essentially a continuation of calculus, and most students who are taking that class probably have a lot of intuition from their experience in calculus, and probably their intuition is correct, and they’re learning how to formalize it. And when you learn about algebraic structures, you probably don’t have very much intuition at all because it’s very abstract.
But students who take topology have intuition because we’re talking about objects that seem familiar, we can draw a lot of pictures, and the intuition is always wrong. It’s very easy to get trapped.
LEVIN: So, what are the techniques that you use to not get trapped?
MILLER: [Laughs] Well, so if I am actually trying to understand objects through pictures, if I’m actually constructing an object visually, then I need to have a rigorous system in place for a picture to actually have meaning. It can’t just be based on my intuition for how the object behaves. I need to formalize if I draw a plane and a line intersecting it, what does that actually mean? And so there’s some real analysis that goes into that.
And so we have to all agree what these diagrams actually mean, and then we also have to develop, here’s rules for what I can do to a picture of a space or a knot inside of a space, and have the picture after I do something to it to change it, have that picture still describe the same space or object or knot. And so once I have done the work, the analysis of understanding these diagrams, that gives me more flexibility to actually do constructions and know that I’m not accidentally losing information.
LEVIN: So, it’s a combination of visual representation, but formalized mathematically?
MILLER: That’s right, and not everything is done through a visual representation. I like to work like that because that’s something that really draws me to mathematics, but there’s lots of excellent topologists, you know, four-dimensional topologists who really don’t make use of visual representation in their work. It’s not always important.
But I do want people to understand, even when math has a lot of pictures in it, you’ll hear this from students a lot, it’s not proof by picture. It’s not just a random picture that we make up. There’s a lot of, that’s right, formalization that goes into these arguments.
LEVIN: With these four-dimensional spaces and your intuition faltering, what are the kinds of questions that you’re asking? What are the hypotheses that you’re really pursuing?
MILLER: So the thing that I study most commonly is the study of knotted surfaces. So in dimension three, the analogous object would be a knot. Knot — k-n-o-t. So a knot is a loop. If I took a piece of string… Wait, I actually, I have a string. If I took a piece of string, and I actually tangled it up, but then I glued the ends together to make one closed loop, then I would call that a knot.
LEVIN: Right.
MILLER: Yeah, and so we, we study those objects up to continuous deformation. So I can stretch the knot around, twist it, but I can never pass it through itself. And this is an important object in three-dimensional topology.
It’s not immediately obvious why it’s important. It takes some theory to understand why knots are important, but it turns out that every three-dimensional space can be understood in terms of knotted loops inside of just classical space.
So in dimension four, I study knotted surfaces, which is everything one dimension up. Instead of a loop, it’s something two-dimensional.
LEVIN: Like a ribbon.
MILLER: Well, less like a ribbon and more like a balloon. So I wanna have an object that doesn’t have any edges. But there’s also more complicated surfaces. So a balloon is a sphere or a 2-sphere. It’s very common to say the dimension of the space.
It’s two-dimensional because if I’m standing on a balloon, then I can go forward, backward, left, right on the balloon, but if I try to go up, I’ll leave the balloon. Maybe instead of a balloon, I should say the surface of the Earth, another example of a two-dimensional sphere. If I go up into space, then I leave the surface of the Earth, so it’s only two-dimensional.
But there are more spaces, like for example, a torus, which is the two-dimensional space that I would get if an asteroid hit the Earth and drilled straight through. So there’s a hole in it now. It’s the surface of that. I think it’d be more normal to say it’s the surface of a bagel or a donut. And then there’s more spaces that I could get if I have more holes.
So I could study surfaces embedded or knotted inside a four-dimensional space. So it’s a little harder to imagine, but there’s some version of tangling, a 2-sphere inside of four-dimensional space.
LEVIN: I have a number of questions. One is you said that you can understand all three-dimensional spaces in terms of these knots.
MILLER: That’s right.
LEVIN: That seems like a big leap. Can you help us make that connection?
MILLER: So this is a big theorem, I think 1960s due to Lickorish and Wallace, Lickorish-Wallace theorem. So, to be more specific, I’m studying three-dimensional manifolds, and I want to say that these three manifolds don’t have boundary, and they’re compact, meaning I don’t ever go out to infinity. So I can, maybe I’ll say finite volume. And I also want them to be orientable. So think about, like, a Möbius band as an example of something that’s not orientable because it has only one side, is what people often say. So, I put some restrictions. I want to only study reasonable three-dimensional spaces.
But once I do that, Lickorish and Wallace proved that I could get to any three-dimensional space I want by starting out in the standard one, which is called the 3-sphere. It’s like R3, but if I try to go out to infinity, then eventually I come back to the point that I started. So I can’t go out for forever.
And all I have to do is take some knots, so some knotted loops inside of the 3-sphere, and I’m going to do surgery on them. And this is confusing. It’s a really important operation in topology. I think it’s surprising the first time you hear it that it could do anything at all. But when I say surgery, I mean I’m gonna delete those knots. So I’m deleting these knotted loops. I wanna think of them as being thick. It’s like I’m cutting out a bunch of bagels from space, so. And then I’m going to just re-glue, like, or replace exactly the thing that I deleted. So I cut out a bunch of these, like, thickened circles, these solid bagels, solid tori. And just reattach them. But I’m going to reattach them in a different way that I found them, and I’ll change the space because of that.
LEVIN: I see. So if I’m understanding correctly, the knots themselves, you were saying I can start with the simply connected finite space, which is the 3-sphere, and I can construct any other topology in three dimensions by performing these kinds of surgeries on knots. Is that what you’re saying?
MILLER: That’s right, and so that’s exactly right. And it’s called Dehn surgery.
LEVIN: And so that way you make more and more complicated spaces far beyond the simplicity of the smooth, compact, connected three-dimensional sphere. You make these very complicated spaces with handles and holes and kind of an origami.
So now take me up to four. Now we’re trying to imagine the role of a knot in four dimensions, but the knots are no longer strings. They themselves are surfaces, complex surfaces,
MILLER: That’s right.
LEVIN: And you’re trying to do the same thing. You’re trying to start with the simplest space in four dimensions and find all the others on the basis of the knots?
MILLER: Yeah. And so it’s almost true now that I can start out with the simplest space, which is called the 4-sphere, which is harder to imagine because it’s four-dimensional. But it’s some standard four-dimensional space that we all agree to start with. And the surfaces that I’m going to take are tori, which are these surfaces of bagels or donuts sitting inside of four-dimensional space. And I’m doing surgery, which means that I’m deleting one at a time. I’m gonna delete a torus, re-glue it back in, but I’m allowed to, like, re-glue it in a funny way, and I’ll change the four-dimensional space.
And so I said it’s almost true that I can get to anything. There are two really basic numerical invariants that I have to start out with the right values. So, at least there is a very simple family of four-dimensional spaces that I start out with. Depending on what space I wanna get, I know which one to start with, and all I have to do is do surgery on tori, and I can get to whatever space I wanted.
LEVIN: I see. And has that been a successful program?
MILLER: Well, so that’s a fact about dimension four. This is a cool theorem, I think it’s of Iwase, several years ago. It’s not exactly a program. It’s more a motivation for why it’s worthwhile to study surfaces. Any question about four-dimensional spaces, in principle, I could rephrase as a question about surfaces, and that’s just one example. There’s other aspects of four-dimensional spaces that we tend to study using surfaces.
So, for example, let’s move back down to a lower dimensional space. Let’s actually look at surfaces again. I gave two examples of surfaces before, which were the 2-sphere and the torus. It’s actually pretty hard the first time you learn about these objects to really verbalize what is the difference between them. When I give a talk, that’s a little bit more general audience, I’ll say something like, “Oh, well, the torus has a hole in it, and the sphere doesn’t have a hole.” But it’s pretty hard on average if I asked a student, “Okay, now tell me what a hole is.” Like, what does it mean that the torus has a hole? That’s a tricky question.
And so the way that I like to think about it is that, well, if I draw a loop on a 2-sphere, so think like the equator of the Earth, it will always divide the 2-sphere into two pieces. The equator divides the Earth into the Northern Hemisphere and the Southern Hemisphere, and you can’t go between them without crossing the equator.
The torus doesn’t have that property, ’cause I could take a loop on the torus that goes around the hole, and if I cut, it actually doesn’t cut the torus into two pieces. But I can only fit one loop like that. If I try to cut along two loops, then I’m gonna end up cutting the torus into two pieces. So, that tells me that the torus has one hole, and I can formalize it in terms of these subspaces, these circles that are smaller in dimension that sit inside of the surface. So, I can do that in higher dimensions, too. If I’m trying to distinguish two 4-manifolds from each other, sometimes I can rephrase that as a question about, “Well, what kind of surfaces sit inside of these four-dimensional spaces?”
[Music plays]
STROGATZ: So when I listen to that with my topologist hat on, I’m thinking that there’s a gambit that Maggie is using here that we’re trained to use in topology all the time. To use, as she puts it, these smaller dimensional objects to probe higher dimensional objects. And since it’s not something we all do every day, I thought rather than talking about the four-dimensional usage, what if we just talked about something like the kitchen table?
LEVIN: Mmm-hmmm.
STROGATZ: You know, so like if I put a piece of string on the table and just make it into the shape of a circle, there’s a very deep topological thing that has happened as soon as I do that, which is that the loop of string separates the table into two parts, one of which we call the inside and one of which we call the outside.
And that’s kind of interesting. I mean, you might say, “No, it’s not. That’s so totally obvious.” But this is the kind of thing a mathematician ponders. Like, okay, let me be more abstract. Rather than a tabletop, how about the infinite plane, you know, the XY plane going out to infinity in both directions?
If you put a circle in the XY plane, it separates the plane into an inside part that’s bounded and an outside part that includes infinity, and that’s true even if I deform the loop, there’s always one inside and one outside.
LEVIN: Yeah. Well, look, we have, skin, two-dimensional surface, and that separates in three dimensions our inside from our outside. But it wouldn’t if we lived in 4D. Our skin would not be sufficient to completely enclose our internal organs.
STROGATZ: Very convenient then that we don’t live in 4D because we’d be spilling.
LEVIN: Right. We’d, we’d just- You’d be able to just, like, go in and pluck out, you know, without surgery.
STROGATZ: Surgery would be a lot easier.
LEVIN: Yeah. Well, she also talks about something that’s similar to your classroom experience, which is in some sense, the connectedness of spaces can be categorized by thinking about how you can divide things, how many cuts you have to make, essentially, to really cut the thing in half.
STROGATZ: Right. The other thing that hit me in her discussion was the difficulty of saying what you mean by a hole. So, you know, where this really became a live discussion on the internet was if I look at a pair of pants, so just ordinary pants, how many holes are in there?
LEVIN: Okay, so there’s one, two.
STROGATZ: But some people wanna say three because, you know, like there’s a loop at the bottom of your leg and then on your other leg and then there’s your belt. It seems like there’s three.
LEVIN: Well, okay. Wait. So, in contractible loops, I have one, two… I guess… three.
STROGATZ: I’m gonna let the audience ponder this one. Where are you taking us next?
LEVIN: Okay. Well, there is something, Steve, that I wanted your help with. There’s a topic that comes up after the break, and that is Seifert surfaces. I think you’ve thought about this.
STROGATZ: I have. They’ve actually come up in some of my applied math work. And probably an easy way to picture them is to imagine a game you might have played as a kid where you take a loop of wire to make bubbles outside. And then if you dunk that in the soapy water, you get a kind of soap film that spans that loop. Well, that disc is an example of a Seifert surface because it’s a surface whose boundary is a given curve, in this case, a loop.
But you could also have Seifert surfaces for knotted curves where the boundary of that surface is the knot. It’s a really useful construction in knot theory.
LEVIN: That’s actually a helpful visualization, and we’re gonna get into that and more just after the break.
[Music plays]
LEVIN: Welcome back to The Joy of Why. We’re joined today by Maggie Miller, a topologist at the University of Texas at Austin.
Let’s talk about this work you did to answer a long-standing question that was posed by Charles Livingston in 1982 with you and your collaborators that received a lot of attention, and that had to do with the Seifert surfaces. What’s special about those surfaces?
MILLER: Yeah, that’s right. And I’ll just say that was with Kyle Hayden, uh, Sungwan Kim, JungHwan Park, and Isaac Sundberg. A Seifert surface is a surface where it’s not just an abstract surface, it’s a surface that actually lives inside of three-dimensional space. So I really wanna make this distinction, even though I know a donut lives inside of our three-dimensional space. I’m treating it as its own abstract object, like, outside of space. It’s in some weird place. But a Seifert surface is really here.
It’s important that they live inside of three-dimensional space because when I study the surface, I’m allowed to continuously deform it, but it has to stay inside of three-dimensional space. So I could have two different tori that are both tori, but I can’t actually turn one into the other inside of three-dimensional space.
A simpler thing is just even knots, knotted loops. If I take a loop that’s really a circle, just a nice round circle, we call that the unknot. If I take a loop and actually tie it into some complicated thing, it won’t be the unknot. And they’re still both circles, but they’re not the same knot. I can’t actually turn one into the other without passing it through itself. Seifert surfaces are the same thing.
But for Seifert surfaces, I do want them to have boundary. They should have an edge, and the edge will be a knot, a knotted loop.
LEVIN: So the idea is you’re imagining, at least with some of the knots, that if you increase the number of dimensions in which they live, that you might be able to unknot them one into the other without cutting or breaking or gluing.
MILLER: That’s right. I mean, that happens very often. So in the case of knots, like a knotted loop, it’s only knotted because we’re in three-dimensional space. If I had a fourth dimension of freedom, if I moved into a four-dimensional space, then my knot would actually become trivial. It’s so small in dimension compared to the ambient space that it can’t be complicated anymore.
LEVIN: So it would untie effectively? It would unknot effectively? Yeah.
MILLER: That’s right. It would just untie. So in, in four dimensions, in the standard four-dimensional space, there is exactly one knotted loop. They’re all the same.
LEVIN: Ah, I see. But that’s not the case if they’re not … Wait, you’re saying if you- if there are knots in three dimensions and you up the dimension to four, then they’re all the same?
MILLER: Then they’re all the same.
LEVIN: But if you have a knot that lives in four dimensions, so it’s a surface knot, and you go up a dimension, are they also then all the same?
MILLER: That’s not so clear, but if I go up two dimensions to dimension six, then they’re all the same. If I increase the dimension enough, eventually they’ll all be the same.
LEVIN: So when I learned a little bit about this, you can take kind of a two-dimensional example, just which might be easier for people to visualize. If you imagine one small rubber band inside a bigger rubber band just on a table. You can’t get them past each other, right?
But if you have three dimensions, it’s real easy. You lift one up, you stretch it out, and then you can lay it outside the other. So this idea of the number of dimensions allowing you to unknot things or to separate things without cutting and pasting is kind of intuitive. We use it all the time.
MILLER: That’s right. That’s exactly the right idea. That’s the same reason as why knots in dimension three become trivial or all the same in dimension four. Like, somehow I just need that one extra dimension to pull the rubber band.
LEVIN: So let’s go back to the Seifert surfaces. What was the question that you addressed about their behavior in four dimensions?
MILLER: Yeah. So we were really interested in this paper by Charles Livingston, Chuck Livingston, where he was studying Seifert surfaces for the unknot and the unlink, where the unlink is a bunch of unknots that are all unlinked from each other. So just a bunch of round circles that have nothing to do with each other. Yeah. So even though the unlink is somehow not very interesting, I could still find complicated Seifert surfaces for the unlink that are different from each other. So the surface boundary, trivial, not interesting. But the surface itself is complicated.
And Livingston proved that if I have two, let’s say, connected Seifert surfaces for the same unlink, and I know that those two surfaces have the same genus, which is our measure of how many holes the surface has.
Even though those surfaces are probably not the same inside of three-dimensional space, if I add a fourth dimension of freedom, then they become the same. I have this extra direction that I can pull them in. In the paper, he was very specific about the boundary being the unlink, and he pointed out that it’s surely not true if the boundary is not the unlink. But he didn’t have a proof of that, primarily because it’s really hard to come up with weird examples of higher-dimensional objects. You have to come up with some crazy pair of surfaces with the same boundary in the first place before you can even then try to prove that they aren’t the same as each other. That’s always the problem in topology.
LEVIN: Yeah. I think we could do a little more justice to the idea of these are the same. That is really one of the core principles of topology is, what do you mean by the same? Can you clarify that?
MILLER: Yeah, it’s a core principle what does it mean for two, uh, objects to be the same. It’s also a core principle that there are lots of different notions of the sameness. So this is always whenever you define an object, you have to say, what is it going to mean for two to be the same. So far we’ve talked about manifolds, which included surfaces and knots. So for manifolds, we typically use the definition homeomorphic or homeomorphism.
LEVIN: Mm-hmm.
MILLER: So we say that two manifolds are homeomorphic if there is a homeomorphism from one to the other. And what that means, if I have my two manifolds, let’s call them X and Y, they’re homeomorphic if I can find a function, F, that goes from X to Y. And the function needs to be continuous, and it also needs to be a bijection, which means that it’s one-to-one and on two.
So, any two points in X have to go to two different points in Y. They can’t go to the same point. And I also have to hit every single point in Y. Something from X has to map to it. And then the last thing is that I want f to have a continuous inverse as well. So it’s just a set of continuous instructions to go from X to Y, but these three bullet points make it a rigorous definition.
So in the context of knots, I can use the same definition, actually. I could say that two knots are equivalent if there’s a homeomorphism, so this kind of restricted function, from 3-space to itself, but when I plug in the first knot, it outputs the second knot.
LEVIN: If I were to give this formalized mathematical description to a child with some Play-Doh, I would say, “The game is, can you make this shape into that shape without breaking it, without puncturing or re-gluing?” As long as you’re just smoothly mushing dough, then topologically we’re gonna consider those homeomorphic.
MILLER: Right. Yeah. So it’s okay to stretch things. It’s okay to twist things. If I have two knots, if I have one knot and I’m trying to turn it into another knot, I can make the string longer and I can add twists, but I can never pass the string through itself, which would break it.
LEVIN: Now, I think the question of games is not totally irrelevant, because for people outside of mathematics, when they’re listening to the problems that mathematicians are addressing, there’s an attempt to assess, is this just a game that’s being played?
And I think the sort of philosophical way to phrase the question is, do you consider mathematics to be something that is discovered, that it exists in some sense out there, or is this or is mathematics sort of a series of different games that are invented by human beings?
MILLER: Oh, that’s interesting. I think, hmm, I think the fundamental or the most important results are things that are discovered, but the way that we get to them is often by winning a game. I think that’s exactly right. If I were trying to solve a problem, let’s say I have a conjecture that I think is true, and I need to prove that it’s true for all knots. Then I might phrase it as a game, well, if I, if I give you a picture of a knot, and I tell you, “Okay, you can start off with this picture, but you’re allowed to change the picture in the following ways,” and I tell you exactly what’s allowed, is it possible to do only those things and eventually get to whatever it is that I’ve conjectured is true?
Like, the part that it’s the discovery, I think, is coming up with the game in the first place. It’s really hard to make a conjecture ‘cause we don’t always have good intuition. And then lots of papers that are just, well, here’s a big problem that we’re all interested in, and my theorem is that I can’t actually answer it, but I can prove that if you can always win this game, then the theorem is true.
LEVIN: For people who are outside of mathematics, who are trying to understand what’s at stake, why are these different games important and compelling? And I know that’s an existential question for mathematics.
MILLER: The games that we were mentioning before are always going into, well, if I could win this game, then I prove this conjecture. So at the heart of it, it’s really why is pure mathematics important? The one issue is that it’s not always clear to anybody working in the field why it’s important for the real world.
People like to joke that there’s, like, a 50-year time gap between pure math being invented or discovered or whichever verb you wanna use versus actually being applied in real life. So for example, in topology, there’s a relatively recent field called topological data analysis, you can tell from the name immediately is much more applicable to real life.
It’s the idea of how can we understand very large data sets efficiently or understand some patterns in them using topology. So maybe I have some data set which is way too massive for me to do a search through, but I can understand global patterns, like, it’s meaningful that somehow this data set I can approximate as this space, and the fact that it’s this space tells me something about the data.
But the actual topology, the pure math part that goes into topological data analysis, is at this point relatively old in terms of just literally the pure math component of it. Maybe we’re using things like classification of surfaces or some classical invariants in topology, and so it’s still somehow catching up. It’s important that we have those things, but the stuff that we’re doing in topology this week isn’t being implemented in applied math yet.
LEVIN: Yeah. And I understand that there’s a sense in which it’s an unfair question. It’s more sort of giving mathematicians an opportunity to address it, because all the time physicists face similar questions. What is the significance of your work to my daily life? And the real answer is that’s not what we care about, and that’s okay.
MILLER: Well yeah, the real answer is that’s not the goal. In fact, I think most topologists, if you ask, they’d probably try to link it back to physics and then hope you don’t ask why the physics part matters.
LEVIN: Right. Well, let’s, you know what? Let’s play that game because that’s interesting and fun. There, there’s a really valuable sense in which topology has huge application to theoretical physics. There’s lots of complex ways in which topology applies, but there’s one that’s kind of intuitive, which is that the universe is probably a three-dimensional manifold. It might be four-dimensional manifold if we include time in the real sense, right? Not just another spatial direction, but time in which measures are different on that manifold for time than they are for space. Do you think about those implications for physics?
MILLER: Well, so actually what’s interesting is that I think most topologists would go the opposite direction and think, “Well, I never think about what physicists need, sorry.” But I’m very interested in what physicists bring to topology.
So when I was in grad school, a professor I was taking a course with made this joke, which I still think about all the time, it’s totally right. Whenever topologists want a new way of distinguishing two spaces from each other, what they do is they walk across the hall to the physics department, which is surely in the same building, and ask them, “Oh, can you please give me a new partial differential equation that tells me something about spaces?” And then you bring it back, and you totally forget that it had anything to do with physics, and that’s just your new way. Yeah, so we, we like to say, “Oh, well, physicists told me that there’s this partial differential equation that describes heat flow, and I can count how many stable solutions there are or whatever, and I’m just gonna remember that number from now on. I’m not gonna think about heat flow, but it’s different for these two spaces, so they must be different.”
LEVIN: I love that. I love that it goes in both directions. I mean, of course, physics also has to grapple with the question of why three dimensions. And so a lot of physicists are interested in higher-dimensional spaces because we think the universe may well be higher dimensional.
And so when, when we’re trying to understand what are the possible universes, what are the geometries of the possible universe, we have to look to the mathematicians and say, “Well, if we’re in four dimensions, these are unique universes in some sense, and these, although they might seem different, are actually the same.” And this is, of course, very relevant for what we’re trying to do.
MILLER: Yeah. That totally makes sense. And of course, there’s lots of mathematicians who study higher dimensional spaces. A lot of people would say, “Oh, I’m a 3-manifold topologist. I’m a 4-manifold topologist.” It’s not so common to have a specific higher number. Like, oh, nobody would say they study 8-dimensional topology…
LEVIN: No? How come? ’Cause it’s too generalizable after eight, it’s just N?
MILLER: They tend to be techniques that are special to three, special to four, and then higher. But there are lots of people who work in higher dimension, especially algebraic topology tends to be higher arbitrary dimension.
LEVIN: Just a curiosity, is there something special about three dimensions and four dimensions?
MILLER: Yeah, there is something special, which I think in dimension three, in some ways it’s similar to dimension two.
In dimension two, surfaces I can enumerate. I can totally list out all of the different two-dimensional objects, and there’s really not very many pieces of data that you would have to give me to totally determine which two-dimensional manifold I have.
In dimension three, it’s a little bit more complicated to state, but there is a complete classification of three-dimensional objects, and this is much more modern. This is going from, maybe ‘70s to 2000s, work of Thurston and Perelman.
We totally… well, maybe I wanna be careful with the word totally, but I would say that we totally understand 3-manifolds. At least we have a complete characterization. There’s still a lot of interesting questions, but it’s generally not are these two spaces different from each other?
In dimension four, that’s a harder question, but there is a lot of interest, not necessarily in a classification. That’s not possible. But it’s special because of the failure of theorems from higher or lower dimension in dimension four, so it’s a little bit different in motivation.
But in higher dimension, we don’t have that anymore, and we also don’t have the classification. And then also there’s this kind of psychological thing of it’s hard enough to imagine a four-dimensional space, but at least with the time analogy, you can kind of get there. But it’s a little bit hard to conceptualize, you know, what’s a 14-dimensional space? So I don’t think there’s that same level of motivation.
LEVIN: We all know that it’s impossible even for the most accomplished mathematician to literally visualize higher dimensional spaces. Is there a sense in which you feel you have a kind of a mental theater?
MILLER: Yeah, I think in dimension four, I really literally imagine and will draw what I described before as a movie. I really like to just draw a sequence of three-dimensional spaces, and so okay, here’s three-dimensional space at time zero, and a little bit to the right of it, I’ll draw another three-dimensional space, be like, this is time one, and a little bit to the left, I’ll draw the same three-dimensional space, be like, this is time negative one. And in some sense, that’s a literal picture of a four-dimensional space. There’s some interpolation of, it’s not really time straight from zero straight to one. There has to be like, there’s one half in the middle, We can’t draw all of them.
You can, in principle, do that in higher dimensions too. I have a paper, maybe a few papers, where I draw five-dimensional spaces as well. This is… I draw four dimensions which are in a row of three-dimensional pictures, so if I stack rows and have a grid, now it’s five-dimensional.
LEVIN: You could go up one more.
MILLER: I have this, like, sort of secret goal of I would really like to draw like a six or maybe seven-dimensional picture where the, one of the dimensions is you turn the page.
LEVIN: Right. Exactly. Or you have to build it in a room.
MILLER: Oh, that’d be interesting too.
LEVIN: I wanted to ask you a question that we like to ask here at The Joy of Why, which is what brings you the most joy in your work?
MILLER: I mean, I think that I have probably the same answer that a lot of mathematicians, where we tend to work on problems that are really hard and abstract, and maybe you think about something for several years before you understand the answer. I was working on more than one thing at a time. I know very few people who are stuck on something for several years, and just that’s the only thing they’re doing. But every once in a while, you actually solve a problem. It feels really good. I mean, I think everybody can relate to that feeling of you’re working on a hard puzzle of some kind, like a crossword, and just this aha moment where you finally get an answer. It’s a big relief.
LEVIN: Yeah. I think it’s interesting to be in a subject where you’re always working at the very edge of your own abilities and maybe what’s even possible in the field. And it can be very frustrating.
MILLER: It’s so frustrating. All the time I just like, “Why can’t I have a hint button?” You know?
LEVIN: [Laughs] Right? Well, Maggie, thank you so much. It’s such an incredible field and I love this connection with the visual. Thank you so much for taking time with us.
MILLER: Yeah, thank you for having me. It was interesting to talk about math.
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STROGATZ: [Laughs] It’s very charming to listen to the two of you talking about what is really a data visualization problem, and I’m reminded of the great book by Edward Tufte, Visual Display of Quantitative Information.
LEVIN: Gripping title.
STROGATZ: Oh, well, okay. Do you know the book?
LEVIN: I don’t, but…
STROGATZ: Oh, come on. You’re teasing me. Okay, fine. The title may not be the selling point for the book, but he has one diagram of Napoleon’s march into Russia. So you see the thickness of a line that represent how many soldiers Napoleon has at a given time. And as this line starts moving across the page, this black band starts getting thinner and thinner as Napoleon’s army is getting depleted. Meanwhile, there’s other data showing the temperature, how cold it was, and what the date was when they attacked this or that place.
So you’re actually getting not only geographical 2D information, you’re getting the size of the army, that’s a third dimension. You’re getting a fourth dimension for the date, and so on. So it’s really clever that people have been thinking about how to visualize multiple dimensions, even on a two-dimensional picture for a long time.
LEVIN: Yeah, visualizations in mathematics is fascinating. I personally really appreciate visualizations, and yet I understand it’s a limitation. But what she described is actually a really fun way to imagine a higher dimensional space, is to unpack it in some sense. Just draw it here, then draw it there.
You know, this idea of a sphere passing through, let’s say, a flat land, a world in which all creatures only live in two dimensions, and what do you see? You can understand the three-dimensional sphere as a series of concentric circles that widen as the sphere enters and then shrink as the sphere leaves.
And it really helps you say, “Well, I could probably do that with a 4-sphere passing through three dimensions,” and indeed you can. It really sort of helps connect with the physicality of these spaces.
STROGATZ: Wow. Aha.
LEVIN: Well, we have to unfortunately leave this abstract realm behind. Back to reality. But Steve, thank you so much for being a partner in crime
STROGATZ: Very good. See you next time.
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STROGATZ: If you’re enjoying The Joy of Why and you’re not already subscribed, hit the subscribe or follow button where you’re listening. You can also leave a review for the show. It helps people find this podcast. Find articles, newsletters, videos, and more at quantamagazine.org.
LEVIN: The Joy of Why is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests or other editorial decisions in this podcast or in Quanta Magazine.
The Joy of Why is produced by PRX Productions; the production team is Caitlin Faulds, Jade Abdul-Malik, Genevieve Sponsler, and Merritt Jacob. The Executive Producer of PRX Productions is Jocelyn Gonzales. Edwin Ochoa is our project manager.
From Quanta Magazine, Simon Frantz and Samir Patel provided editorial guidance, with support from Samuel Velasco, Kit Sudol, Simone Barr, and Michael Kanyongolo. Samir Patel is Quanta’s Editor-in-Chief.
The episode art is by Chanelle Nibbelink and our logo is by Jaki King and Kristina Armitage. Special thanks to Garth Avery at the Cornell Broadcast Studio.
I’m your host, Janna Levin. If you have any questions or comments, please email us at [email protected]. Thanks for listening.
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